Waves & Oscillations PYQ

phy-1109 PHY-1109 Physics

OSCILLATIONS AND MECHANICAL WAVES

Subtopic A: Simple Harmonic Motion (SHM) Fundamentals

Question	Year(s) of Appearance
Defination: simple harmonic motion	2016, 2017, 2018, 2021 2022, 2023, 2024
Explain simple harmonic motion and discuss its oscillatory behavior.	2016, 2023
Show graphical representation of the SHM.	2017, 2022
Derive the differential equation for SHM and show that its solution is of the form $y(t)=A\sin (\omega t+\varphi )$.	2017, 2024, 2024(becm)
Show in tabular form and graphically that simple harmonic motion is an oscillatory motion in terms of displacement velocity and acceleration.	2018, 2021, 2022
Discuss some basic characteristics of Simple Harmonic Motion.	2024(becm)

Subtopic B: SHM Energy, Superposition, and Calculations

Question	Year(s) of Appearance
Show that the total energy of a particle executing simple harmonic motion is constant.	2016
Two SHM’s acting simultaneously on a particle are given by the equation. $Y_1=2\sin (\omega t+\pi /4)$ and $Y_2=3\sin (\omega t+\pi /3)$. Calculate (i) amplitude (ii) phase constant (iii) time period of the resultant vibration.	2016
Two SHM $y_1=5\sin (\omega t+\pi /2)$, $y_2=6\sin (\omega t+\pi /3)$. Determine (i) amplitude, (ii) phase constant, (iii) time period of the resultant vibration. Construct the equation.	2024(becm), 2024
What is superposition principle? Show that superposition principle is valid only in case of linear homogeneous equation.	2018, 2019, 2020, 2021, 2024(becm)
Derive an expression for energy of vibration of a vibrating system.	2021
A particle is moving simple harmonically… velocities $u_1$ and $u_2$ at $x_1$ and $x_2$… show that the period is given by $T=2\pi \sqrt{(x_1^2-x_2^2)/(u_2^2-u_1^2)}$. Find also its maximum velocity and amplitude.	2017
A SHM is presented by $y=12\sin (6t-\pi /3)$. Calculate the frequency, epoch, maximum displacement, velocity and acceleration at $t=0$ and $t=6$ seconds.	2018
Body vibrating with SHM of amplitude 15 cm and frequency 4 Hz. Determine maximum velocity/acceleration/speed when body is 9 cm from equilibrium.	2021
Force $F=5\sin (\pi t)$, displacement $y=0.50\sin (\pi t-\pi /4)$. Calculate the work done in 30 seconds and in 5 minutes.	2019
Force $F=25\sin (\pi t/10)$, displacement $Y=0.5\sin (\pi t/10-\pi /4)$. Calculate the work done by the excitation force in (i) 30 seconds and (ii) 5 minutes.	2020
Force $F=0.5\sin (2t)$, displacement $Y=0.5\sin (2t-\pi /2)$. Calculate the work done by the excitation force in (i) 12 minutes and (ii) one day.	2022
What are Lissajous figures? Derive an expression for two simple harmonic vibrations of equal time periods acting at right angles. Discuss when the phase difference $\varphi =0,\pi /2,\pi ,\pi /4,$ and $3\pi /4$.	2022, 2024, 2020
Draw the Lissajous’ figure using the following equations: $x=a\sin \omega t$ and $y=2a\sin 2\omega t$.	2017
Two SHM $x=a\sin (\omega t+\varphi )$, $y=b\sin (\omega t)$. Obtain the general equation of resultant vibration. Draw the Lissajous figures for $\varphi =\pi /2$ and $2\pi$.	2018

Subtopic C: Damped and Forced Oscillations / Resonance

Question	Year(s) of Appearance
Discuss the sharpness of resonance.	2016
What are damped vibrations and forced vibrations? Derive an expression for power dissipation in damped harmonic oscillation.	2017, 2020
What is resonance? Derive an expression for maximum displacement of a driven oscillator.	2017, 2020
Under damped harmonic oscillator: amplitude reduced to 1/10th after 100 oscillations. T=1.25 sec. Calculate (i) damping constant and relaxation time, (ii) amplitude without damping.	2017, 2020
What is Damped oscillation? Write forces, construct and solve the differential equation, obtain critical damping condition.	2018
What is damped vibration? Derive and solve the differential equation of damped simple harmonic motion in LCR circuit.	2019, 2021
Define bandwidth of resonance. Derive an expression for bandwidth of resonance.	2019
Deduce the frequency and quality factor for a circuit with $L=2$ mH, $C=50\mu F$, and $R=2$ ohms.	2019
Derive and solve the differential equation of damped vibration in LCR circuit. Discuss the case when damping is small.	2021
Particle mass 2 gm, damping force $8\text{ dyne-sec/cm}$. Periodic force $250\sin (2t)$ dyne applied. Determine frequency for displacement resonance and velocity resonance.	2021
Obtain an expression for the displacement in the case of a LCR circuit. Discuss the effect of critical damping on the LCR circuit.	2023
Find whether the discharge of a condenser for the following inductive circuit is oscillatory: $C=0.1\mu F,L=10$ mH, $R=210$ ohms. If the circuit is oscillatory, calculate its frequency.	2023
Distinguish between free, forced and damped vibrations.	2024(becm)
Harmonic oscillator Q=12 subjected to sinusoidal force. (i) amplitude of forced oscillation (ii) angle $\varphi$ by which it will be out of phase with the driving force.	2024(becm)
Describe the term “Quality factor”. Derive an expression for the energy loss per cycle in a damped harmonic oscillator.	2024
Construct an expression for the displacement in the case of a damped oscillatory motion. Also find out the maximum displacement of a driven oscillator in case of resonance.	2024
Calculate the frequency and quality factor for a circuit with $L=5.2$ mH, $C=5\mu F$, and $R=0.35$ ohm.	2024

Subtopic D: Wave Motion (Progressive, Stationary, Group/Phase Velocity)

Question	Year(s) of Appearance
Show that the energy of a plane progressive wave is $E=2{\pi }^2\rho n^2{a}^2$.	2016, 2021, 2023
What is meant by intensity of a plane progressive wave? Obtain an expression for the intensity of a plane progressive wave.	2024(becm)
What is wave velocity? Derive an expression for energy of a progressive wave.	2018
Wave along a string $y=0.02\sin (30t-4x)$. Find amplitude, frequency, speed and wavelength.	2018
What are stationary waves? Write down their properties.	2017, 2021
What is stationary wave? Discuss analytically the formation of stationary waves when reflection occurs at a free boundary. Why are they referred to as stationary waves?	2019
What are nodes and anti-nodes? Discuss the formation of stationary waves due to reflection at a free end boundary. Explain how characteristics change (i) with position and (ii) with time.	2024(becm)
Express mathematically that no energy is transfer in case of stationary wave.	2024
Stationary wave $y=0.8\sin (0.32x)\cos (16t)$. What are amplitude and velocity of individual waves?	2019
Define group velocity and phase velocity. Find a relation between them.	2023
What is phase velocity and group velocity?	2021, 2022
Mention the basic difference between phase velocity and group velocity. Establish the mathematical relation between them.	2024(becm)
Show that the group velocity depends on the medium through which the wave is transmitted.	2019, 2022

Subtopic E: Acoustics and Sound

Question	Year(s) of Appearance
What is Doppler’s effect in Sound? Obtain an expression for the apparent frequency of a note when the source and the listener are (i) moving towards each other, (ii) moving away from each other.	2016
What is Doppler’s effect? Derive an expression for the apparent frequency of a note when (i) The source and observer is moving separately towards the stationary observer and source. (ii) The source and the listener are moving.	2018
What is Doppler’s effect in sound? Obtain an expression for the apparent frequency of a note when the source and the listener are (i) moving towards each other and (ii) moving away from each other.	2022
Briefly explain the Doppler principle.	2017
Two aeroplanes pass each other… calculate the frequencies of the notes heard in the other aeroplane (i) before and (ii) after they have passed each other.	2016
Apparent frequency of whistle changes in ratio 6:5 as engine passes stationary observer. Calculate $V_{engine}$.	2022
What is Beats? Show that the number of beats produced per second is $f_1-f_2$.	2018, 2022
Define about beats. How are they produced? Analytically describe the formation of beats and prove that the number of beats per second is equal to the difference in frequency of the two notes.	2024(becm), 2024
What is bel and phone? Discuss the factors influencing loudness.	2016, 2024(becm)
Calculate the increase in the acoustic intensity level when the sound is doubled.	2017
Calculate the change in intensity level when the intensity of sound increases 100 times its original intensity.	2023
Calculate the (i) acoustic intensity (ii) acoustic pressure of a plane, acoustic wave in air of intensity level of 80 decibels reference to $10^{-12}{\text{ watt/m}}^2$.	2016, 2022
Calculate acoustic intensity and acoustic pressure of a plane acoustic wave in air of intensity level of 100 dB reference to $10^{-12}{\text{ watt/m}}^2$.	2021
Sound source $f=250$ Hz, $A=0.40$ cm. Determine rate of flow of energy per square cm.	2024(becm)
Sound source $f=256$ Hz, $A=5\times 10^{-5}$ m. Calculate the flow of energy across a square meter per second.	2024
Explain Sabine’s reverberation formula.	2016, 2022
Derive an expression for the growth and decay of sound in an auditorium and hence obtain Sabine’s reverberation formula.	2023, 2024(becm)
Write down the requisites for a good auditorium.	2017, 2023
Room volume 750 m${}^3$. Calculate average absorption coefficient and reverberation time given coefficients for walls, floor, and ceiling.	2024(becm)
Room $6\times 5\times 4$ meters. Calculate (i) mean free path of the sound wave, (ii) number of reflection made per second by the sound wave.	2023